- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Corrections of the GR eikonal limit by a class of renormalizable gravity models

Leandro Lanosa* and Osvaldo P. Santillán†

  • *Contact author: lanosalf@hotmail.com
  • †Contact author: firenzecita@hotmail.com
PDF

Phys. Rev. D 113, 044012 – Published 4 February, 2026

DOI: https://doi.org/10.1103/61tk-bzts

Abstract

In the present work, the scattering between a light scalar particle ϕ and a heavy scalar σ in the eikonal limit is considered, for gravity scenarios containing higher order derivatives. It is suggested that if one of the new gravity scales introduced in the higher order action is smaller than the Planck mass, for instance of the order of MGUT∼1015  GeV, the functional form of the GR eikonal formulas appears changed by a factor. However, in this situation, the conditions for the eikonal approximation to hold has to be revised; this issue is analyzed in the text. The statements of the present work should be taken with a grain of salt, as the Schwarzschild radius for these polynomials theories is not yet established. The results presented here, in our opinion, are in agreement with the suppression of corrections to GR pointed out in other literature regarding Stelle gravity. The next-to-leading-order approximation and part of the seagull diagrams are estimated. Different to the GR case, this order generically is nonvanishing. An explicit regularization scheme is presented, based on Riesz and Hadamard procedures. The need of a regularization is partially expected, as the inclusion of small energy fluctuations may spoil the eikonal approximation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (132)

  1. K. Stelle, Phys. Rev. D 16, 953 (1977).
  2. K. Stelle, Gen. Relativ. Gravit. 9, 353 (1978).
  3. M. Asorey, J. López, and I. Shapiro, Int. J. Mod. Phys. A 12, 5711 (1997).
  4. L. Modesto and I. Shapiro, Phys. Lett. B 755, 279 (2016).
  5. L. Modesto, Nucl. Phys. B909, 584 (2016).
  6. L. Modesto, T. de Paula Netto, and I. Shapiro, J. High Energy Phys. 04 (2015) 098.
  7. N. Burzilla, B. Giacchini, T. de Paula Netto, and L. Modesto, Phys. Rev. D 103, 064080 (2021).
  8. B. DeWitt, Phys. Rev. 162, 1195 (1967).
  9. P. Mannheim, Found. Phys. 37, 532 (2006).
  10. P. Mannheim and A. Davidson, Phys. Rev. A 71, 042110 (2004).
  11. C. Bender, Rep. Prog. Phys. 70, 947 (2007).
  12. A. Smilga, Phys. Lett. B 632, 433 (2005).
  13. K. Andrzejewski, J. Gonera, and P. Maslanka, Prog. Theor. Phys. 125, 247 (2009).
  14. R. Di Criscienzo and S. Zerbini, J. Math. Phys. (N.Y.) 50, 103517 (2009); 51, 059901(E) (2010).
  15. S. Ketov, G. Michiaki, and T. Yumibayashi, arXiv:1110.1155.
  16. T. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, Phys. Rev. Lett. 108, 031101 (2012).
  17. T. Chen and E. A. Lim, J. Cosmol. Astropart. Phys. 05 (2014) 010.
  18. B. Holdom, Phys. Lett. B 843, 138023 (2023).
  19. K. Andrzejewski, J. Gonera, P. Machalski, and K. Bolonek-Lason, Phys. Lett. B 706, 427 (2012).
  20. A. Salvio, A. Strumia, and H. Veermae, Eur. Phys. J. C 78, 842 (2018).
  21. A. Salvio, Front. Phys. 6, 77 (2018).
  22. A. Salvio and A. Strumia, Eur. Phys. J. C 76, 227 (2016).
  23. J. Donoghue and G. Menezes, Phys. Rev. D 100, 105006 (2019).
  24. M. Levy and J. Sucher, Phys. Rev. 186, 1656 (1969).
  25. S. Wallace, Ann. Phys. (N.Y.) 78, 190 (1973).
  26. J. Cardy, Nucl. Phys. B28, 477 (1971).
  27. M. Abarbanel and C. Itzykson, Phys. Rev. Lett. 23, 53 (1969).
  28. H. Cheng and T. Wu, Phys. Rev. 186, 1611 (1969).
  29. V. A. Matveev and A. N. Tavkhelidze, Teor. Mat. Fiz. 9, 44 (1971).
  30. B. Barbashov and V. Nesterenko, Teor. Mat. Fiz. 4, 293 (1970).
  31. B. Ioffe, V. Fadin, and L. Lipatov, Quantum Chromodynamics: Perturbative and Nonperturbative Aspects (Cambridge University Press, Cambridge, England, 2010), ISBN [Amazon][WorldCat].
  32. Y. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy (Cambridge University Press, Cambridge, England, 2012), ISBN [Amazon][WorldCat].
  33. V. Barone and E. Predazzi, High-Energy Particle Diffraction (Springer, New York, 2002), ISBN: [Amazon][WorldCat].
  34. H. Verlinde and E. Verlinde, Nucl. Phys. B371, 246 (1992).
  35. G. t Hooft, Phys. Lett. B 198, 61 (1987).
  36. G. t Hooft, Nucl. Phys. B304, 867 (1988).
  37. D. Kabat and M. Ortiz, Nucl. Phys. B388, 570 (1992).
  38. L. Lipatov, Phys. Part. Nucl. 44, 391 (2013).
  39. M. Duff, Phys. Rev. D 7, 2317 (1973).
  40. D. Kosower, B. Maybee, and D. O’Connell, J. High Energy Phys. 02 (2019) 137.
  41. B. Holstein and J. Donoghue, Phys. Rev. Lett. 93, 201602 (2004).
  42. C. Cheung, I. Rothstein, and M. Solon, Phys. Rev. Lett. 121, 251101 (2018).
  43. Z. Bern, J. Parra-Martinez, R. Roiban, E. Sawyer, and C. Shen, J. High Energy Phys. 05 (2021) 188.
  44. Z. Bern, J. Parra-Martinez, R. Roiban, M. Ruf, and C. Shen, Phys. Rev. Lett. 126, 171601 (2021).
  45. N. Bjerrum-Bohr, J. Donoghue, B. Holstein, L. Plante, and P. Vanhove, J. High Energy Phys. 11 (2016) 117.
  46. C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. Porto, Phys. Rev. Lett. 130, 101401 (2023).
  47. G. Kalin and R. Porto, J. High Energy Phys. 11 (2020) 106.
  48. G. Kalin, Z. Liu, and R. Porto, Phys. Rev. Lett. 125, 261103 (2020).
  49. C. Diappa, G. Kalin, Z. Liu, and R. Porto, Phys. Rev. Lett. 128, 161104 (2023).
  50. C. Diappa, G. Kalin, Z. Liu, and R. Porto, Phys. Rev. Lett. 132, 22 (2024).
  51. C. Diappa, G. Kalin, Z. Liu, and R. Porto, J. High Energy Phys. 08 (2023) 109.
  52. S. Foffa and R. Sturani, Classical Quantum Gravity 31, 043001 (2014).
  53. I. Rothstein, Gen. Relativ. Gravit. 46, 1726 (2014).
  54. R. Porto, Phys. Rep. 633, 1 (2016).
  55. M. Levi, Rep. Prog. Phys. 83, 075901 (2020).
  56. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Rep. 1083, 1 (2024).
  57. G. Dvali, C. Gomez R. Isermann, D. Lüst, and S. Stieberger, Nucl. Phys. B893, 187 (2015).
  58. D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 197, 81 (1987).
  59. D. Gross and P. Mende, Phys. Lett. B 197, 129 (1987).
  60. D. Amati, M. Ciafaloni, and G. Veneziano, Int. J. Mod. Phys. A 03, 1615 (1988).
  61. D. Gross and P. Mende, Nucl. Phys. B303, 407 (1988).
  62. D. Amati, M. Ciafaloni, and G. Veneziano, Nucl. Phys. B347, 550 (1990).
  63. D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 289, 87 (1992).
  64. D. Amati, M. Ciafaloni, and G. Veneziano, Nucl. Phys. B403, 707 (1993).
  65. M. Ciafaloni, D. Colferai, and G. Veneziano, Phys. Rev. D 99, 066008 (2019).
  66. M. Ciafaloni and D. Colferai, J. High Energy Phys. 10 (2014) 085.
  67. I. Aref’eva, Colliding hadrons as cosmic membranes and possible signatures of lost momentum, in Cosmology, Quantum Vacuum and Zeta Functions, edited by S. Odintsov, D. Sáez-Gómez, and S. Xambo-Descamps, Springer Proceedings in Physics Vol. 137 (Springer, Berlin, Heidelberg, 2011).
  68. M. Ciafaloni, D. Colferai, F. Coradeschi, and G. Veneziano, Phys. Rev. D 93, 044052 (2016).
  69. J. Muzinich and M. Soldate, Phys. Rev. D 37, 359 (1988).
  70. Y. Iwazaki, Lett. Nuovo Cimento 1S2, 783 (1971).
  71. N. Gaddam and N. Groenenboom, J. High Energy Phys. 01 (2022) 146.
  72. D. Amati, M. Ciafaloni, and G. Veneziano, J. High Energy Phys. 02 (2008) 049.
  73. M. Ciafaloni, D. Colferai, and G. Falcioni, J. High Energy Phys. 09 (2011) 044.
  74. M. Ciafaloni and D. Colferai, Phys. Rev. D 96, 126008 (2017).
  75. N. Gaddam and N. Groenenboom, Phys. Rev. D 109, 026007 (2024).
  76. N. Gaddam, N. Groenenboom, and G. t’Hooft, J. High Energy Phys. 01 (2022) 023.
  77. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Lett. B 818, 136379 (2021).
  78. G. D’Appollonio, P. Di Vecchia, R. Russo, and G. Veneziano, J. High Energy Phys. 11 (2010) 100.
  79. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, J. High Energy Phys. 07 (2021) 169.
  80. A. Koemans Collado, P. Di Vecchia, and R. Russo, Phys. Rev. D 100, 066028 (2019).
  81. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Lett. B 811, 135924 (2020).
  82. S. B. Giddings, D. J. Gross, and A. Maharana, Phys. Rev. D 77, 046001 (2008).
  83. C. Heissenberg, Phys. Rev. D 104, 046016 (2021).
  84. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, J. High Energy Phys. 07 (2022) 039.
  85. P. Di Vecchia, C. Heissenberg, and R. Russo, J. High Energy Phys. 08 (2022) 172.
  86. A. Georgoudis, C. Heissenberg, and R. Russo, J. High Energy Phys. 03 (2024) 089.
  87. B. Bellazzini, G. Isabella, and M. Riva, J. High Energy Phys. 04 (2023) 023.
  88. A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, J. High Energy Phys. 06 (2023) 126.
  89. G. Mogull, J. Plefka, and J. Steinhoff, J. High Energy Phys. 02 (2021) 048.
  90. B. Holstein and J. Donoghue, Phys. Rev. Lett. 93, 201602 (2004).
  91. A. Luna, I. Nicholson, D. O’Connell, and C. White, J. High Energy Phys. 03 (2018) 044.
  92. N. Bjerrum-Bohr, P. Damgaard, G. Festuccia, L. Planté, and P. Vanhove, Phys. Rev. Lett. 121, 171601 (2018).
  93. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Lett. B 843, 138049 (2023).
  94. X. Camanho, J. Edelstein, J. Maldacena, and A. Zhiboedov, J. High Energy Phys. 02 (2016) 020.
  95. M. Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, Phys. Rev. D 101, 046014 (2020).
  96. K. Hinterbichler, A. Joyce, and R. Rosen, J. High Energy Phys. 03 (2018) 051.
  97. C. Chen, C. de Rham, A. Margalit, and A. Tolley, Phys. Rev. D 111, 024066 (2025).
  98. F. Serra, J. Serra, E. Trincherini, and L. Trombetta, J. High Energy Phys. 08 (2022) 157.
  99. K. Haring and A. Zhiboedov, Sci. Post Phys. 16, 034 (2024).
  100. H. Lu, C. Pope, A. Perkins, and K. Stelle, Phys. Rev. Lett. 114, 171601 (2015).
  101. H. Lu, C. Pope, and K. Stelle, Phys. Rev. D 92, 124019 (2015).
  102. H. Lu, C. Pope, and K. Stelle, Int. J. Mod. Phys. A 30, 1545016 (2015).
  103. A. Fazio and E. Reyes, Mod. Phys. Lett. A 36, 2150138 (2021).
  104. R. Saotome and R. Akhoury, Phys. Rev. D 84, 104040 (2011).
  105. R. Akhoury, R. Saotome, and G. Sterman, Phys. Rev. D 103, 064036 (2021).
  106. T. Biswas, T. Koivisto, and A. Mazumdar, arXiv:1302.0532.
  107. A. Bartoli, J. Julve, and E. Sanchez, Classical Quantum Gravity 16, 2283 (1999).
  108. Boris N. Latosh, https://github.com/BorisNLatosh/FeynGrav.
  109. B. Latosh, Classical Quantum Gravity 39, 165006 (2022).
  110. B. Latosh, Comput. Phys. Commun. 292, 108871 (2023).
  111. B. Latosh, Comput. Phys. Commun. 310, 109508 (2025).
  112. D. Prinz, Classical Quantum Gravity 38, 215003 (2021).
  113. E. Fradkin, Nucl. Phys. 76, 3588 (1966).
  114. E. S. Fradkin, V. Esposito, and S. Termini, Riv. Nuovo Cimento 2, 498 (1970).
  115. H. M. Fried, Green’s Functions and Ordered Exponentials (Cambridge University Press, Cambridge, England, 2002).
  116. M. Alves, R. Cuzinatto, C. de Melo, L. Medeiros, and P. Pompeia, Phys. Rev. D 111, 084055 (2025).
  117. M. F. S. Alves, L. F. M. A. M. Reis, and L. G. Medeiros, Phys. Rev. D 107, 044017 (2023).
  118. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, edited by Alan Jeffrey and Daniel Zwillinger, Sixth Edition (Academic Press, Amsterdam, 2015).
  119. Eugene P. Wigner, Phys. Rev. 98, 145 (1955).
  120. A. P. Martin, Acta Phys. Aust. Suppl. 23, 157 (1981).
  121. A. A. Tseytlin, Phys. Lett. B 176, 92 (1986).
  122. S. Deser and A. N. Redlich, Phys. Lett. B 176, 350 (1986); 186, 461(E) (1987).
  123. M. Ruhdorfer, J. Serra, and A. Weiler, J. High Energy Phys. 05 (2020) 083.
  124. A. Salvio, J. Cosmol. Astropart. Phys. 07 (2024) 092.
  125. A. Salvio, Phys. Lett. B 856, 138920 (2024).
  126. S. Paycha, Regularised Integrals, Sums, and Traces: An Analytic Point of View (American Mathematical Society, Providence, 2012).
  127. R. Estrada and R. Kanwal, J. Math. Anal. Appl. 141, 195 (1989).
  128. I. Gelfand and G. Shilov, Generalized Functions (Academic Press, New York, 1964), Vol. 1.
  129. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, England, 1944), ISBN [Amazon][WorldCat].
  130. Tables of Integral Transforms, edited by H. Bateman and A. Erdélyi (McGraw-Hill Book Company, 1954).
  131. Anatolij P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, and G. G. Gould, Integrals and Series Vol. 3—More Special Functions (Gordon and Breach Science Publishers, New York, 1990), English Version, ISBN [Amazon][WorldCat].
  132. N. Suan Han, Do Thu Ha, and Nguyen Nhu Xuan, Eur. Phys. J. C 79, 835 (2019).

Outline

Information

Phys. Rev. D 113, 044012– Published 4 February, 2026
  • Received 6 May 2025
  • Accepted 12 January 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation